DEGREE PROJECT, IN CIVIL AND ARCHITECTURAL ENGINEERING , SECOND LEVEL
STOCKHOLM, SWEDEN 2015
Dynamic Analysis of Long Span Footbridges
DYNAMIC ANALYSIS OF FOOTBRIDGE IN PROJECT SLUSSEN
YINA FAN& FANGZHOU LIU
KTH ROYAL INSTITUTE OF TECHNOLOGY
TRITA BKN. Master Thesis 452 ISSN 1103-4297
ISRN KTH/BKN/EX-452-SE
www.kth.se
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Preface
The$ work$ associated$ with$ this$ master$ thesis$ was$ carried$ out$ at$ the$ Department$ of$ Civil$ and$
Architectural+Engineering,+Royal+Institute+of+Technology,+KTH+University,)Sweden)from)January) to#June#2015.!
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We# would# like# to# express# our! deepest& gratitude& to& our& supervisors,& Frank& K& E& Axhag& at& ELU&
Company(and(Professor(Costin(Pacoste"Calmanovici*at*KTH*for*their*professional*guidance*and*
encouragement+ during+ the+ entire" work" of" this" master" thesis." We" also" owe" gratitude" to" Ph.D."
Christoffer*Svedholm!for$supporting$us$with$ideas$and$patience.!
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Lastly,(we(would(also(like(to(thank(the(staff(and(our(fellow(students(at(the(Department(of(Civil(
and$Architectural$Engineering$for!their&help&and&useful&conversation&during&coffee&breaks&and&
leisure'time.
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Yina%Fan%&%Fangzhou%Liu!
May!2015!
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Abstract
A footbridge in Slussen is planned to be built and will connect the area of Gamla Stan with Sodermalm. As an increasing number of footbridges with large span tend to become more flexible and light these days, the corresponding dynamic problems due to decreased stiffness and mass draw much more attention. Specifically speaking, reduced stiffness and mass lead to smaller natural frequencies, which make the structure more sensitive to pedestrian-induced loading, especially in lateral direction.
Fortunately, in this master thesis, only the vibration in vertical direction is focused due to that the footbridge in Slussen project uses enough lateral bracings to make sure that the safety of lateral vibration is kept at an acceptable level.
In order to analyze dynamic response of the footbridge, the real footbridge structure is converted into a FE model by the commercial software LUSAS. In this thesis, four different kinds of critical standards are introduced, which are Sétra [8], Swedish standard Bro 2004 [9], ISO 10137 [5] and Eurocode respectively. By comparing these four criteria, Sétra and Eurocode are finally chosen to be the standard and guidelines for this project. They give the basic theories about how to model the pedestrian loading and provide critical values to check the accelerations in both vertical and lateral direction.
By using FE software LUSAS, natural frequencies of the footbridge and the corresponding mode shapes can be calculated directly. Then, according to these results and relevant theories introduced by Sétra, the pedestrian loading can be modeled and the acceleration response of any specific mode can be obtained as well.
Finally, based on the worst case with excessive acceleration, the methods to reduce dynamic response will be presented. Commonly, there are two ways to reduce acceleration response. One method is to increase the stiffness of the structure. However, the increased stiffness is always accompanied with increased mass of the structure. Because of this reason, the other way that installing dampers is widely used in recent years. In this thesis, the tuned mass dampers (TMDs) are introduced in detail as well as the information about the design principles of it. With important parameters known, TMDs can be added to the model to check how the accelerations can be reduced.
Key words: footbridge, pedestrian-induced vibration, Sétra, Eurocode, vertical acceleration vibration, TMD damper system.
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Concents!
1 INTRODUCTION!...!1!
1.1! BACKGROUND!...!1!
1.2! AIMS AND SCOPE!...!1!
1.3! METHOD!...!2!
1.4! DISPOSITION!...!2!
2 BASIC THEORY!...!3!
2.1! STRUCTURAL DYNAMICS!...!3!
2.1.1! SDOF system!...!3!
2.1.2! MDOF system!...!6!
2.2! DYNAMIC LOAD INDUCED BY PEDESTRIANS!...!8!
2.2.1! Walking load!...!8!
2.2.2! Pedestrian running load!...!11!
3 STATE-OF-ART IN LONG SPAN FOOTBRIDGE!...!13!
3.1! SOLFÉRINO FOOTBRIDGE!...!13!
3.1.1! Introduction!...!13!
3.1.2! Design!...!14!
3.1.3! Method to reduce dynamic response!...!14!
3.2! 1STHUI KETING BRIDGE IN MIAN YANG!...!16!
3.2.1! Introduction!...!16!
3.2.2! Dynamic analysis!...!17!
3.2.3! Method to reduce dynamic response!...!17!
3.3! MILLENNIUM BRIDGE!...!18!
4 STANDARDS AND GUIDELINES!...!21!
4.1! SÉTRA!...!21!
4.2! BRO 2004!...!22!
4.3! ISO10137!...!22!
4.4! EUROCODE!...!24!
5 THE FOOTBRIDGE!...!25!
5.1! STRUCTURE OF THE FOOTBRIDGE!...!25!
5.2! FE MODEL!...!26!
5.2.1! Geometry!...!26!
5.2.2! Mesh!...!27!
5.2.3! Material!...!29!
5.2.4! Support and foundation!...!31!
5.2.5! Model checking!...!31!
6 DYNAMIC ANALYSES OF FOOTBRIDGE!...!33!
6.1! CALCULATION OF NATURAL FREQUENCIES AND MODE SHAPES!...!33!
6.2! DYNAMIC RESPONSE INDUCED BY PEDESTRIAN WALKING!...!34
6.2.1! Design load!...!34!
6.2.2! Dynamic response in terms of accelerations!...!36!
7 IMPROVING DYNAMIC BEHAVIOR!...!40!
7.1! INCREASING STIFFNESS!...!40!
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7.2! INSTALLING DAMPERS!...!41!
7.2.1! Introduction of TMD!...!42!
7.2.2! Design principle of TMD!...!43!
7.2.3! Results after installation of TMDs!...!45!
8 CONCLUSIONS!...!51!
8.1! SUMMARY!...!51!
8.2! CONCLUSIONS!...!51!
8.3! FUTURE WORK!...!52!
APPENDIX A!...!61!
APPENDIX B!...!63!
APPENDIX C!...!67!
APPENDIX D!...!69!
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Chapter 1
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Introduction
1.1 Background
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A footbridge in Slussen is planned to be built and will connect the area of Gamla Stan with Sodermalm for the sake of convenience for pedestrians and bicyclists. Looking back to past years, with the construction technology becoming increasingly mature and the usage of high strength construction materials, the structure of footbridges becomes more flexible and lighter.
Due to the application of large slenderness and low mass, long span footbridges are particularly sensitive to dynamic effects induced by streams of pedestrians crossing the bridge. The excessive vibrations of resonance response caused by consistency of frequency of pedestrian loads and natural frequencies of footbridge need to be paid more attention. By taking this into consideration, designers normally concentrate on analyzing the vertical vibration. However, after what happened to London Millennium Bridge (LMB) due to synchronization and lock-in effects, it is turned out that the transversal vibrations can also create major structural damage and affect the safety of footbridges. Therefore, in Slussen project, according to dynamic response analysis, both vertical and transversal vibrations should be checked and compared with regulatory requirements.
1.2 Aims and scope
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The aim of this master thesis is to analyze the dynamic performance of the long span footbridge in project Slussen and to find solutions to reduce excessive vibrations. Only pedestrian loading is considered in this master thesis. Special attention is given to the dynamic response of structures induced by pedestrian loading in vertical direction. As for the dynamic performance in transversal direction, the synchronization and risk of lock-in caused by a crowd of pedestrians also need to be studied. The work can mainly be divided into three subtasks:
! A literature state-of-the-art review in long-span footbridges.
! Dynamic analyses of the footbridge in Slussen. The natural frequencies of the structure and accelerations in both vertical and lateral direction should be checked.
! Present methods to reduce the dynamic performance. The priority is given to the installation of dampers.
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2! ! CHAPTER(1.(INTRODUCTION!
1.3 Method
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To begin with, a literature study of standards, guidelines, and how to model the dynamic loading induced by pedestrian walking and jogging is performed. This provides the basic foundation for further analyses.
After the literature review, the modeling of the footbridge itself is required. The model of the bridge shall be built by using FE software LUSAS based on the geometry that is already known.
LUSAS is a commercial software used for Finite Element (FE) analysis. It should be noted that some simplifications and assumptions are made when using LUSAS to model the structure.
Once the model of the structure is finished, the pedestrian loads will be applied to the bridge in order to analyze the dynamic response calculated as accelerations. The numerical analysis of pedestrian loading can be carried out with the assistance of MATLAB, which is not within the scope of this thesis. If the accelerations do not fulfill the requirements, some solutions, for instance, adding dampers shall be considered. The design methodology of dampers and how the dampers will be added in our model require specific attention.
1.4 Disposition
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This master thesis mainly consists of 8 chapters. It starts with a literature study of theories about dynamic problems that is covered by Chapter 2. Chapter 2 also introduces theories of dynamic loads induced by pedestrian walking and running.
In Chapter 3, a literature review of state-of-art in long span footbridge is presented. Some examples of designs of worldwide footbridges with respect to reducing dynamic response are discussed in this Chapter. Chapter 4 provides a brief introduction of different standards and guidelines dealing with dynamic analyses of footbridges.
The FE model of the footbridge is built by FE software LUSAS. The content about how to build the model and some important aspects with respect to it are given by Chapter 5. The dynamic analyses of the Slussen footbridge are performed in Chapter 6. This chapter presents the modes which suffers from dynamic problems, which sets a foundation for further study associated with taking measures to reduce dynamic responses.
Chapter 7 discusses different solutions to solve excessive vibrations introduced in Chapter 6. The comparison between different solutions is discussed in Chapter 7 and finally, the most optimal solution is given.
The conclusions and future work are summarized in Chapter 8.
Finally, some detailed calculations and Matlab codes are provided in Appendix.
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Chapter 2
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Basic theory
2.1 Structural dynamics
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Dynamic problems are associated with analyzing the response of structures under dynamic loads.
The major characteristic of dynamic loads is that they are time-dependent. Therefore, the response (e.g. displacement, velocity, and acceleration) is a function of time. The objective of structural dynamics is to develop the equation for them and then solve the equations.
A dynamic model can either be a basic or a more complicated model. The basic model is a system with a single degree of freedom system (SDOF), which means only one coordinate is used to define the motion of the system. A more complicated one is a system with multi degree of freedom (MDOF). The analysis of this type of problem is based on that of a single degree of freedom system. In the following sections, both of these systems will be discussed in details.
Firstly, the equation of motion for each system will be developed. Then, different analysis methods to solve the equation of motion are presented.
2.1.1 SDOF system
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In this part, the theory of a generalized SDOF system is described. The primary objective is to formulate the equation of motion, then, the method of numerical evaluation of dynamic response is used for solving the equations.
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Equation of motion
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The mathematical expressions defining the dynamic displacements are called the equation of motion of the structure and the solution to these equations provides the required displacement time histories. The equation of motion of any dynamic system represents the expression of Newton’s second law, which states that the rate of change of momentum of any mass particle is proportional to the force acting on it. The equation of motion for a generalized SDOF system is
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!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! + !! + !" = ! ! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!(2.1)!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
! Where M is the mass of the structure, C is the damping of the structure, K is the stiffness of the structure, and ! ! is the external force [1].
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Analysis methods
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In numerical evaluation of dynamic response, the method of numerical time-stepping integration for the equation of motion is generally used. When using this approach, three important requirements for a numerical procedure should be considered: (1) convergence—as the time step decreases, the numerical solution should approach the exact solution, (2) stability—the numerical solution should be stable in the presence of numerical round-off errors, and (3) accuracy—the numerical procedure should provide results that are close enough to the exact solution [1]. Here, only two commonly used methods are explained below.
The first method is known as central difference method (also called explicit method). This method is based on a finite difference approximation of the derivatives of displacement (velocity and acceleration). By taking constant time steps, ∆!! = ∆!, the central difference expressions for velocity and acceleration at time i are:
!!! = !!!!− !!!!
2∆! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!(2.2)!
!! =!!!! − 2!!+ !!!!
(∆!)! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!(2.3)!
Substituting Eq.2.2 and Eq.2.3 for velocity and acceleration in Eq. 2.1 gives:
!!!!!− 2!! + !!!!
(∆!)! + !!!!!− !!!!
2∆! + !!! = !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!(2.4)!
In Eq.2.4, the displacements !!!and !!!! are assumed known from implementation of the procedure for the preceding time steps. Transferring these known quantities to the right side leads to:
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∆! !+ !
2∆! !!!! = !! − !
∆! ! − !
2∆! !!!! − ! − 2!
∆! ! !!!!!!!!!!!!!!!!!!!!!!!(2.5)!
or
!!!!!! = !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!(2.6)!
where
! = !
∆! !+ !
2∆!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!(2.7)!
and
!! = !!− !
∆! !− !
2∆! !!!!− ! − 2!
∆! ! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!(2.8)!
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The displacement at time !!!! can be solved by the equilibrium at time !!, which is typical for explicit methods [1]. The explicit method depends on primary displacement and is very sensitive, compared with the implicit method introduced later.
Another method is Newmark’s method (also called implicit method). In 1959,N.M.Newmark developed a family of time-stepping methods based on the following equations:
!!!! = !! + 1 − ! ∆! !!+ !∆! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!(2.9)!!!!!!!!!!!!!!!!!!!!!
!!!!! = !! + ∆! !! + 0.5 − ! ∆! ! !!+ ! ∆! ! !!!!!!!!!!!!!!!!!!!(2.10)!!!!!!
The parameters ! and ! define the variation of acceleration over a time step and determine the stability and accuracy characteristics of the method [1]. Besides, it should be noted that the prerequisite of using Newmark’s method is to make sure the structure system is a linear system.
For nonlinear system, Newmark’s original formulation can be modified in order to get results from Eq.2.9 and Eq.2.10 without iteration. The equation of motion becomes:
!!!!! + !!!!! + !!!!! = !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!(2.11)!
In Eq.2.10,!!!!! can be expressed in terms of !!!!:
!!!! = ! ∆!! ! !!!!− !! −!∆!! !! − (!!! − 1)!!!!!!!!!!!!!!!!!!!!!!!!!!(2.12)!
Substituting Eq.2.12 into Eq.2.9 gives:
!!!! = !∆!! !!!! − !! + 1 −!! !!+ ∆!(1 −!!!)!!!!!!!!!!!!!!!!!!!!!!(2.13)!
Eq.2.12 and Eq.2.13 can be substituted into Eq.2.11 at time i+1 to obtain:
!!!!!! = !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!(2.14)!
where
! = ! + !
!∆!! + 1
! ∆! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!(2.15)!
and
!!!! = !!!!+ 1
! ∆! !! + !
!∆!! !! + 1
!∆!! + !
!− 1 ! !!!
!!+ 1
2!− 1 ! + ∆! !
2!− 1 ! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!(2.16)!
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! and !!!! can be calculated from the structure properties( m, k, and c). Parameters !, !, and the state of the system at time i can be presented by !!,!!!, and !!. Finally, the displacement at time i+1 is calculated from:
!!!! = !!!!
! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!(2.17)!
If !!!! is known, the velocity !!!! and acceleration !!!! can be determined from Eq.2.12 and Eq.2.13 respectively.
In Newmark’s method, the solution at time i+1 is determined from Eq.2.11, the equilibrium condition at time i+1. The characteristic of this method is always stable for any ∆!, however, the more accurate result is sensitive to time step ∆! that can be determined by equation:
∆! = 1
40!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!(2.18) where f is the natural frequency of the structure.
2.1.2 MDOF system
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The structure having more than one degree of freedom (DOF) is defined as multi-degree-of freedom (MDOF) system. In reality, almost all the structures have infinite number of DOF. In order to solve the problems of MDOF system, the Finite Element (FE) analysis is a necessity. To be more specific, the structure can be idealized as an assemblage of elements interconnected at nodes. The displacements of the nodes are degree of freedom. The mass of the structure is lumped at nodes and the stiffness of members can be approximated for linear system.
This section presents the equation of motion for the MDOF system and the methods to solve it.
There are two methods: modal analysis methods and direct integration methods. The method of modal analysis will be explained later.
Equation of motion
The equation of motion for MDOF system is similar to that of SDOF system except that the stiffness and damping are expressed as matrix form. The stiffness matrix of the whole structure can be obtained by assembling the stiffness matrix of single element. The damping matrix is also calculated in the same way. The equation of motion is:
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!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! + !! + !! = ! ! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!(2.19)!!!
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Where M is the mass matrix of the structure, C is the damping matrix and K is the stiffness matrix. All of them are N*N matrix where N is the number of DOF of the whole structure. This
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is a system of N ordinary differential equations governing the displacements u(t) due to applied forces p(t) [1].
Analysis methods
As mentioned above, the displacement u in Eq.2.1 is a function of time and can be solved by either modal analysis method or direct integration method. Here, only the modal analysis is presented.
The modal analysis is associated with the free vibration of the structure. By free vibration we mean the motion of a structure without any dynamic excitation-external forces or support motion [1]. Free vibration is initiated by disturbing the system by some initial displacement and then let the system vibrate freely. By using the modal analysis, the natural frequency and the corresponding vibration mode can be determined. The natural frequency is a frequency at which the structure tends to vibrate while its mode shape is the deflected shape occurred at this specific frequency. They can be solved as typical eigenvalue problems. It should be noted that one mode shape corresponds to one natural frequency.
For the free vibration of un-damped system in one of its natural vibration modes, the displacement is described by:
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! ! = !!(!)!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!(2.20)!
Where !! is the deflected shape. The time-dependent factor !!(!) for each mode is described by simple harmonic function:
!!!!!!!!!!!!!!!!!!!!!!!!!! ! = !!!"#!!! + !!!"#!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!(2.21)!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
Where !! is the natural frequency, !! and !! are constants that are determined by the initial condition of structure. Combining Eq.2.20 and Eq.2.21 will give us:
! ! = ! !! ! (!!!"#(!!!) + !!sin!(!!!))!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!(2.22)!
Substituting u(t) into Eq.2.19 with C and p(t) equal to zero gives:
! − !!!! !! = !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!(2.23) This equation will have a nontrivial solution if:
!!!!!!!!!!!!!!!!!!!!!!!!!!!det ! − !!!! = 0!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!(2.24)!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
Eq.2.24 is known as the characteristic equation or frequency equation [1]. Due to the stiffness matrix K and mas matrix M are symmetrical and positive, this equation has N real and positive roots for !!!.
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Once the mass and stiffness matrices for the structure are determined, natural frequencies and corresponding mode shapes can be solved by eigen-command function in Matlab.
After having determined !! ! !and!ϕ!, the displacement can be calculated as the sum of modal contributions:
!!!!!!!!!!!! ! = !!!!+!!!!!+ ⋯ + !!!! = !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!(2.25)!!!!!!!!!!!!!!!!!!!!!!!
2.2 Dynamic load induced by pedestrians
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The bridge in project Slussen is a pedestrian bridge, thus this master thesis only focuses on the loads from foot traffic. The following sections discuss the load induced by both pedestrian walking and jogging. Jumping is not considered in this case. The load is decomposed into 3 directions: vertical, longitudinal, and lateral direction. Only the vertical component and lateral component are considered. The vertical load of one person from foot traffic can be expressed by Fourier expression [2]:
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! ! = ! + !!!!!!!sin 2!"!!! − !! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!(2.26)!!!!!!!!!!!!!!!!!!!!!!
Where G is the self-weight of one person that can be assumed to be 700N; !! is Fourier coefficient for harmonic component; !! is the loading frequency, and !! is the phase angle. The parameters in Eq.2.26 are given different values depending on the load type (walking or jogging) and load direction (vertical or transversal).
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2.2.1 Walking load
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The characteristic of walking load is that, when walking, there is always one foot in contact with the ground. As what was mentioned above, the load consists of vertical component and lateral component that should be analyzed separately.
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Load in vertical direction
For normal walking with pacing frequency !! = 2!", the vertical component of one foot has a saddle shape as shown in Fig.2.1. The first peak corresponds to the impact of heel on the ground and second peak corresponds to thrust of heel on the ground.
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Figure'2.1:'Force'resulting'from!walking[12]!
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One of the most important parameters to define the walking load in vertical direction is the vertical forcing frequency. Generally, for normal walking, the vertical loading frequency is described as a Gaussian distribution with a mean value of 2Hz and standard deviation of 0.173Hz [3]. This result is based on experiment performed by Matsumoto et al [3], where the walking frequencies of 505 people are measured and summarized in Fig.2.2.
Figure 2.2: Vertical loading frequency for normal walking [3]
According to experimental results, the vertical loading frequency is mainly in the range of 1.6- 2.4Hz [4]. This frequency range is almost the same with that proposed by ISO 10137 [5] ranging from 1.2 to 2.4Hz.
As for the Fourier coefficients !!, different guidelines will give different suggestions. For instance, Blanchard et al. [6] proposed a moving force by only taking the first term in Fourier series into account with !! = 0.257. Another suggestion was proposed by Reiner et al. [7], based on which, the first four terms in Fourier series need to be considered and the coefficient for each term is frequency dependent, see Fig.2.3.
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Figure 2.3: !!!till !!for normal walking [7]
ISO 10137 [5] gives !! = 0.37 ! − 1 if f=1.2-2.4Hz, !! = 0.1, and !! = 0.06 . It also stipulates the phase angle ! is equal to !/2 if frequency is below the resonance frequency. For the design in serviceability limit state, the suggestion in Sétra [8] is adopted. Specifically speaking, if the vertical eigenfrequency is in the range of 1-2.6Hz, only the first harmonic term with !! = 0.4 is considered since the resonance will occur at first harmonic. If the eigenfrequency is in the interval of 2.6-5Hz, only the second harmonic term with !! = 0.1 is considered since the resonance will occur at second harmonic.
Load in transversal direction
The amplitude of dynamic response of the structure in transversal direction is relatively small compared with that in vertical direction. However, the lock-in effect has been limited to transversal footbridge vibrations, which makes load in transversal direction a significant aspect to study. The lock-in phenomenon refers to the tendency to synchronize the pacing rate with the motion of bridge. The in-situ tests on Solférino footbridge and the Millennium footbridge confirm that the pedestrian modifies his walking pace when the transverse motion of footbridge begins to disturb him [8].
The Eq.2.8 can also be used to describe transversal load, however, it should be noted that the loading frequency in transversal direction is half of the frequency of vertical load. As a result, the frequency of transversal load lies in the region of 0.6-1.2 Hz.
The Fourier coefficients in this case are selected based on Sétra [8]. For eigenfrequency of 0.3- 1.3Hz, only the first harmonic term with !! = 0.05 is needed. For eigenfrequency of 1.3-2.5 Hz, only the second harmonic term with !! = 0.01 is considered.
Load from crowd
Practically, footbridges are subjected to load from a crowd of pedestrians. The behavior of several pedestrians or crowd is very complicated due to several reasons. To begin with, due to their different entrance time, there is always a phase shift between pedestrians. Besides, there is more or less synchronization between pedestrians. Apart from that, pedestrians will modify their behaviors according to the motion of bridge, which is almost impossible to simulate in software.
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In order to simplify the simulation, the assumption that the load from a crowd of people can be obtained by multiplying the load from one pedestrian by a factor is made. According to Matsumoto et al [3], the force generated by N persons walking on the bridge is defined as:
! ! = !!! ! = ! ! + !!!sin 2!"!!! − !!
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!!!!!!!!!!!!!!!!!!!!!!!!!!(2.27)!
A basic assumption for Eq.2.27 is that the maximum density of pedestrians on the bridge is 0.1 person per square meter. In Bro 2004[9], a value of 0.1 pers/m2 is assumed. The Fourier series is limited to the first term with G=600N and !! = 0.257 [6].
Another alternative is provided by Sétra [8]:
!!!!! ! = !!"!! ! = ! !!" ! + !!!sin 2!"!!! − !!
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!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!(2.28)!
Where !!" refers to the number of pedestrians who are perfectly synchronized. For sparse crowd, !!" = 10.8 !", where N is the number of pedestrians and ! is the critical damping ratio. For very dense crowd, !!" = 1.85 ! [8]. The Fourier coefficient given by Sétra is already mentioned above.
When modeling in LUSAS, the crowd load is modeled as uniformly distributed load per square meter. As shown in Fig.2.4, the uniformly distributed load must be given an amplitude sign that is exactly the same as the mode shape sign. In this case, the frequency of the load is set to be equal to the natural frequency of the structure in order to obtain the maximum acceleration at corresponding resonance.
Figure 2.4: Distribution of the crowd load according to mode shape sign.
2.2.2 Pedestrian running load
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In this section, only the vertical vibration induced by pedestrian running is explained due to the transversal vibration is not sensitive to pedestrian running. Apart from that, the phenomenon of synchronization and lock-in hardly appear since if the transversal vibration is manifest, the people on bridge cannot keep running, instead, they will tend to walk or stand statically.
According to ISO 10137[5], the allowance frequency range for running is 2.0-4.0Hz.
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Fourier transform can still represent for the running load. In order to make allowance for the discontinuous contact with ground when running, only the positive part of the transform is retained which may be written, with the previous notation [8]:
!! ! = !! + !!!!"#
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2!"!!! − Φ! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!(2.29)!
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Where !!=700N is the weight of single pedestrian, ! is Fourier coefficient,!!! is vertical frequency and Φ! is phase angle. In this case, Fourier coefficients can be selected according to Reiners [7], see Fig.2.5, based on experiment data.
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Figure 2.5: Reiner[7] !! to !! for running
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Furthermore, the ISO 10137 [5] gives the following value for the Fourier coefficients: !! = 1.4,
!! = 0.4, and!!! = 0.1. It is noted that the values of the Fourier coefficients according to ISO 10137 is consistent with Reiner’s proposal. For designing the running load, the proposal from Grundmann et al [11] is used, which is presented as:
!!" = !!! ! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!(2.30)!
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Where S is synchronization factor. This proposal is valid for small groups (max 10 persons). It is based on the assumption that all pedestrians have the same velocity but different frequencies and step length. Grundmann [11] proposed a multiplication factor as shown in Fig.2.6. Although this is normally used for walking, this factor can also be applied for running.
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Figure 2.6: Multiplication factor for small groups (max 10 persons)[11]
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Chapter 3
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State-of-art in Long span Footbridge
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This chapter provides a literature review of worldwide long span footbridges. Three footbridges are introduced in this chapter. They are Solférino Footbridge in France, 1st Hui Keting Bridge in China, and Millennium Bridge in London. For each footbridge, we focus on the conceptual idea behind the design of the structure and methods to reduce dynamic response of them.
3.1 Solférino footbridge
The Solférino footbridge in France, also known as Passarelle Leopold-Sedar-Senghor, was opened in 1999 to link Musee d’Orsay to Tuileries Gardens. Unfortunately, on the opening day, unexpected lateral oscillation occurred and the bridge was closed until 2000. This section includes a brief introduction of Solférino footbridge as well as the design principle. Finally, the method to solve the problem is described in details.
3.1.1 Introduction
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The Solférino footbridge is designed as a steel arch bridge which is made up of two arching forms connected at their crowns, see Fig.3.1. The footbridge has only one span and the length of the bridge is approximately 140m. The structure consists of a welded steel arch substructure and a timber arch deck supported by the substructure. These two members are connected by V- shaped trusses. The footbridge is driven by concepts of lightweight and slenderness. When walking on the bridge, pedestrians should feel carried by voids underneath them. Besides, the design of footbridge also allows pedestrians to walk along struts on lower stairs. In order to make the structure lightweight, the struts are split in a V-shape and there are no diagonal members used [18]. From Fig.3.1, it is obvious that the lower arch has a much smaller radius than the upper one, which enables the space to decrease toward their crowns. The footbridge has a large number of struts that enable the load to act as uniformly distributed load in order to make good use of the arch form.
Figure 3.1: Solférino footbridge
14! CHAPTER(3.(STATE"OF"ART$IN$LONG$SPAN$FOOTBRIDGE!
3.1.2 Design
!
The most important structural element of Solférino footbridge is the lower arch which is curved in two planes. Each plane is divided into two ribs that splay out at the center and reconnect at the abutment [18]. The truss is in the form of so-called virendeel truss that contains no diagonal element. Besides, every third vertical member of the truss is made bigger than the rest to make a contribution of structural stiffness.
The loads from the bridge deck are transferred through V-shaped struts lined up with stiffer members of truss, see Fig.3.2. The connection is obtained with specifically shaped braces that were built to encompass struts completely. The bridge deck is made from tropical wood and the rest of the structure are made from steel S355 [19].
Figure 3.2: Approach from lower part
3.1.3 Method to reduce dynamic response
!
Since Solférino footbridge experienced dynamic problems, some measures were taken in order to improve its comfort level. After dynamic analyses, the first eight vibration modes of Solférino footbridge was calculated and the corresponding mode shapes are shown in Fig.3.3. Additionally, the on-situ test which discovered the first ten vibration modes with frequency less than 5Hz was performed based on empty bridge with extremely small damping ratio. After further analyses, three of ten vibration modes are considered critical: the first lateral mode coupled with torsional movement at 0.81Hz and two torsional modes at 1.94Hz and 2.22Hz respectively [20]. The results of first six vibration modes are listed in Table 3.1.
Figure 3.3: First eight vibration mode [19]
3.1.$SOLFERIO$FOOTBRIDGE! 15!
Table 3.1. Results of on-situ test [20]
Mode Description Natural
frequency (Hz)
1 Lateral 0.81
2 Vertical bending 1.22
3 Torsion 1.59
4 Vertical bending 1.69
5 Torsion 1.94
6 Torsion 2.22
In February 2000, a series of crowd tests including 122 pedestrians was performed and the results indicated that there was large lateral response of the first mode with an acceleration up to 0.6m/s2. In order to reduce the dynamic response, the installation of TMDs was selected. As a result, 14 TMDs were installed, 6 of which were used for suppressing the lateral vibration and the rest were used for reducing vertical vibration [18].
Practically, there are several types of TMDs available. The most typical one consists of a mass connected to the structure by means of vertical coil springs and one or more hydraulic or pneumatic dampers [8]. When the horizontal vibrations need to be damped, a damper consisting of a mass attached to the bottom of a pendulum should be adopted. The detailed properties of dampers for Solférino footbridge are given in Table 3.2 and the layout of them is shown in Fig.3.4.
Table 3.2. Properties of dampers
Properties of dampers
Vertical TMD Double mass spring system:
2×2 masses 2500 kg at mode of 1.94Hz
2×2 masses 1900 kg at mode of 2.22Hz
Horizontal viscous damper
Pendulum on oil:
6 masses 2500kg at mode 0.8Hz
Figure 3.4: Cross section with dampers
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3.2 1
stHui Keting Bridge in Mian Yang
The 1st Hui Keting Bridge in Mian Yang, China, was opened in 2012 and was the first double- layer cable-stayed bridge that separates pedestrians and vehicles, see Fig.3.5. This section includes a short introduction of this bridge, dynamic response induced by pedestrian loading, and method to improve the dynamic performance.
Figure 3.5: The view of 1st Hui Keting bridge in Mian Yang, China
3.2.1 Introduction
!
The total length of 1st Hui Keting Bridge in Mian Yang, China, is approximately 1400m. The length of main bridge is 400m and the main span has a combination of 100m+200m+100m. The width of main bridge is 28m. Both left and right pylons are designed as inverted "Y" shaped towers and the total height of main tower is 98m. The bottom layer of the bridge is used for vehicles while the upper layer with the length of 681.363m and the width of 6m is designed as footbridge for pedestrians. The shape of the upper footbridge is the “S” type. The main girder of footbridge is a typical steel box girder as shown in Fig.3.6.
3.2.$1ST!HUI$KETING$BRIDGE$IN$MIAN$YANG$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$17!
Figure 3.6: The main girder of footbridge
The stayed cables are designed as single cable plane and the total number of cables is 68 (52 cables are used for vehicle bridge while others are used for pedestrian bridge).
3.2.2 Dynamic analysis
!
The natural frequencies of the bridge and corresponding mode shapes were calculated by ANSYS which is also a commercial FE software. The characteristic of the bridge is its curved shape and due to this feature, the vertical vibration is always accompanied by transversal vibration. It is significantly important to reduce both vertical vibration and horizontal vibration.
The results of accelerations exceeding the comfort criteria are shown in Table 3.3 below.
Table 3.3. Accelerations exceeding requirements
Mode number Frequency (Hz) Maximum vertical acceleration (m/s2)
Maximum transversal acceleration (m/s2)
2 0.4588 0.079 0.256
5 0.7314 0.208 0.113
14 1.620 1.323 1.165
15 1.678 0.84 1.463
18 1.960 2.776 0.641
20 2.410 2.037 0.631
3.2.3 Method to reduce dynamic response
!
According to what was obtained from dynamic analyses, some measures must be taken in order to improve the comfort level of the footbridge. By comparing different methods, the designer finally chose to add tuned mass dampers (TMDs) to reduce accelerations. There were 6 sets of TMDs corresponding to 6 vibration modes existing problems. The parameters of TMDs are listed in Table 3.4 below. After the TMDs were installed, the dynamic behavior was dramatically improved, which could satisfy the requirements.
18! CHAPTER(3.(STATE"OF"ART$IN$LONG$SPAN$FOOTBRIDGE!
Table 3.4. Parameters of TMDs
Mode Location of TMD
Motion of TMD
Damper mass (t)
Spring stiffness (kN/m)
Damping coefficient
(kNs/m)
2 mid-span transversal 33.41 249.25 26.59
5 -43m&51m transversal 12.35 252.18 9.57
14 168m vertical 13.77 1302.55 35.8
15 168m transversal 11.08 1099.56 36.16
18 -62.5m&50m vertical 1.86 278.59 2.77
20 -126.5m&73.5m vertical 8.74 1946.7 22.38
(Note:'the'coordinates'of'left'pylon'and'right'pylon'are'"100m$and$100m$respectively)!
3.3 Millennium Bridge
!
The London Millennium Bridge shown in Fig.3.7 is located across the Thames River in Central London. This bridge is opened in June 2000 in order to celebrate the coming of the new century.
However, during the first day there were a large number of pedestrians crossing the bridge, which made the maximum crowd density reach 1.3-1.5 per square meter at any one time [24]. As a result, the Millennium Bridge suffered from excessive lateral vibration caused by a synchronized-effect of horizontal pedestrian loading. The bridge had to be closed after the opening day. After the failure of Millennium Bridge, international engineers and researchers have been paying much more attention to the study of excessive lateral vibration.
Figure 3.7: The London Millennium Bridge
3.3.#MILLENNIUM#BRIDGE! ! 19!
!
The London Millennium Bridge is a suspension bridge with three spans. The length is 144m for main span, 81m for north span, and 104 m for south span. The suspension system of the bridge consists of the supporting cables below the deck level, which gives a very shallow profile [25].
The bridge has two piles. The aluminum bridge deck is 4m wide and supported by a fabricated steel box section spanning between the two cable groups every 8 meters, see Fig.3.8. The 8 suspension cables (4 suspension cables as a cable group on each side of deck) are tensioned to pull with a force of 2,000 tons against the piers set into each bank. It is enough to support a pedestrian load of 5,000 people on the bridge at one time.
Figure 3.8: The cross section of London Millennium Bridge
The vertical damping system of the London Millennium Bridge was provided by tuned mass dampers (TMDs). Even though the vertical vibration did not experience any problem, a total number of 26 pairs of vertical TMDs were installed in the London Millennium Bridge. The TMDs consist of masses between 1000 and 3000kg supported on compression springs and located on top of the transverse arms beneath the deck [26], see Fig.3.9.
20! CHAPTER(3.(STATE"OF"ART$IN$LONG$SPAN$FOOTBIRDGE!
Figure 3.9: Vertical TMDs
The lateral damping system of the bridge was provided by viscous dampers that can add energy dissipation to the bridge structure. A fluid viscous damper dissipates energy by pushing fluid through an orifice to produce a damping pressure. The basic structure of the fluid viscous damper is shown in Fig.3.10 [27]. In order to reduce the excessive lateral vibration, 37 fluid viscous dampers were installed on the London Millennium Bridge, see Fig.3.11. After the installation of dampers, the maximum lateral acceleration was reduced to 40 times of previous one [28].
Figure 3.10: Basic structure of fluid viscous damper
Figure 3.11: The layout of fluid viscous dampers
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21!
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Chapter 4
!
Standards and Guidelines
!
This Chapter includes widely used standards and guidelines associated with dynamic analyses of footbridges. The requirements with regard to either natural frequencies or vibration amplitudes are explained below. It should be noted that the vibration amplitude should be checked only if the eigenfrequency criteria cannot be fulfilled. The criteria regarding to vibration amplitudes are given in terms of accelerations which reflects a comfort level of the bridge. All of the following standards give the definition of acceptable accelerations. As for the natural frequencies, the relevant requirement can be found in Eurocode.
4.1 Sétra
!
The comfort requirements proposed by Sétra [8] are developed by the technical department for Transport, Roads and Bridge engineering and Road safety in France. For vertical vibrations, Sétra defines the accelerations by using four intervals according to the degree of comfort level.
Definitions of different comfort levels and corresponding vertical acceleration intervals are listed in Table 4.1. The choice of comfort level highly depends on the users, especially the sensitive populations (children, elderly or disabled people). Besides, the significance of footbridge also plays an important role in determining the comfort level.
Table 4.1.#Definition(of(comfort(level(and(associated(accelerations([8]!
Comfort level
Definition Acceleration intervals
m/s2 High comfort The vibrations are imperceptible for persons on the
bridge.
!!,!"#! < 0.5
Mean comfort
The vibrations are merely perceptible for persons on the bridge.
0.5 < !!,!"#! < 1
Low comfort The vibrations of the bridge are clearly perceptible but
not intolerable. 1 < !!,!"#! < 2.5
Unacceptable !!,!"#!> 2.5
In addition to that, the transversal acceleration in any case should be limited to 0.1m/s2 based on experimental data from Solférino footbridge test. However, this value has nothing to do with comfort level but decides the risk of lock-in effect.
22! CHAPTER(4.(STANDARDS(AND(GUIDELINES!
4.2 Bro 2004
!
Bro 2004 [9] published by the Swedish Road Administration (SRA) used to be the general technical standard applied to the design and construction of bridges in Sweden.
In Bro 2004 [9], it is stated that for footbridges affected by pedestrian loads, the acceptable natural frequency of vertical vibration should be larger than 3.5 Hz. Besides, it requires that the acceleration in vertical direction should not exceed 0.5m/s!. However, this requirement refers to the r.m.s.acceleration instead of maximum acceleration. The a!"# can be calculated as following expression:
!!"# !! = 1
! !(!)!
!!!!
!!
!"!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!(4.1)!
!
!The unknown τ is calculated as a function of natural frequency of first bending mode:
! = 1
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!(4.2)!
!
Furthermore, Bro 2004 states a simplified method that is used for calculating vertical accelerations for dynamic analyses. The vertical acceleration should be calculated by making assumption that the pedestrian load is modeled as a harmonic vertical concentrated stationary force F. This force should be regarded to provide the great resultant vertical vibration acceleration [9]:
! = !!!!sin 2!!!! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!(4.3)!
!
Where k! = 0.1BL and k!=150 N are loading constants, !! is the frequency of the load, t is the time, B is the width of the bridge and L is the length of the bridge between supports. Here, it is noted that Bro 2004 only introduces the vertical vibration. There is no comment with respect to horizontal vibration.
4.3 ISO 10137
!
ISO 10137 [5] was prepared by Technical Committee ISO/TC 98, Bases for design of structures, Subcommittee SC 2, Reliability of structures. It provides the basic standards and guidelines for the design of structures (bridges and buildings) with respect to vibrations in serviceability limit state. In Annex C, the comfort requirement that takes the operating frequency into account can be found. These requirements are shown as reference curves shown in Fig.4.1 and Fig.4.2, which show the relationship between the permissible acceleration and the frequency.
!
!
!
4.3.$ISO$10137! ! 23!
!
!
! !
Figure 4.1: Reference curve for vertical vibration
!
! !
Figure 4.2: Reference curve for transversal vibration
!
24! CHAPTER(4.(STANDARDS(AND(GUIDELINES!
The above requirements also relate to the r.m.s. acceleration calculated by using Eq.4.1. A suggestion value for!τ is 1s [5].
4.4 Eurocode
!
EN 1990: 2002 A1
!
EN 1990- Basis of Structural design [13] defines the concept of pedestrian comfort criteria in two ways. Firstly, it notes that the pedestrian comfort criteria should be performed only if the natural frequency of the bridge deck is less than [13]:
! 5 Hz for vertical vibrations
! 2.5 Hz for horizontal (lateral) and torsional vibrations
Secondly, the pedestrian comfort criteria should be defined in terms of maximum acceptable acceleration of any part of the bridge deck. The recommended maximum values are given [13]:
! 0.7 !/!! for vertical vibration
! 0.2 !/!! for horizontal vibrations due to normal use
! 0.4 !/!! for exceptional crowd conditions EN 1991-Part 2
!
In EN 1991-Part 2: Traffic loads on bridges [14] states a recommended method to model pedestrian loads according to the dynamic characteristics of the structure (the natural frequencies of the main structure of the bridge deck). To be more specific, the forces exerted by pedestrians with frequency that is identical to natural frequency of the bridge can result in resonance and need to be taken into account for limit state verification in relation with vibrations [14].
Furthermore, there is also a definition of the frequency range for walking pace of pedestrians.
The frequency ranges are given as [14]:
! In the vertical direction, with a frequency range of between 1 and 3 Hz
! In the horizontal direction, with a frequency range of between 0.5 and 1.5 Hz
! Groups of joggers may cross a footbridge with a frequency of 3 Hz EN 1995-Part 2
!
In EN 1995-Part 2: bridges [15], there is an introduction regarding to design of timber bridges. In this part, it describes in details how to calculate the accelerations in both vertical and horizontal vibrations induced by one person, several persons and joggers respectively. Besides, the recommended maximum value of acceleration mentioned in EN1990 can also be applied in timber bridges.
!
!
25!
Chapter 5
!
The Footbridge
In this chapter, the work on how to build the model of the footbridge in Slussen project is discussed in detail. The commercial software LUSAS is used for building the FE model and performing dynamic analyses of the footbridge. In order to check the accuracy of the model, a convergence test and a comparison of LUSAS results with hand calculations are performed.
5.1 Structure of the footbridge
!
The footbridge in project Slussen is designed as a continuous beam in 6 spans. It will be built close to the existing subway bridge. In order to simplify the analyses, only the straight part of the footbridge is studied. The total length of the bridge is approximately 217m. The length of each span is shown in Fig.5.1.
Figure 5.1: Elevation of footbridge
The cross section of the footbridge is already given in Fig.5.2. The bridge deck is supported by two I-shaped girders and covered by asphalt pavement with mean thickness of 0.1m. There are two edge beams located at each side of the bridge respectively. Apart from that, parapets on each side of the footbridge should be considered and included in the model of the footbridge. The lateral bracings should be paid specific attention as well. The aim of installing them is to increase the lateral stiffness, which in turn, can reduce the dynamic response in lateral direction.
Normally, the stiffness of lateral bracings at support is larger than that in mid span.
Figure 5.2: Cross section of footbridge
26! ! CHAPTER(5.(THE(FOOTBRIDGE!
The abutment of the footbridge is supported by pile foundation as shown in Fig.5.3. However, the foundation of the footbridge needs not to be included in the FE model. What is required is the realistic support condition of the structure.
Figure 5.3: Pile foundation of footbridge
5.2 FE model
!
In order to analyze the dynamic response induced by pedestrians, a 3-dimentional finite element (FE) model of the structure was built. In this thesis, the FE software LUSAS was chosen as a tool to perform FE analyses. Generally, the analysis process in LUSAS consists of three steps. In the first step, a model of the structure should be established. Then, the simulation process where LUSAS solves the numerical problems is performed based on the established model. The third step is post processing which will provide us the results in a form we desire.
In the following sections, the major aspects of LUSAS model as well as a model checking are discussed in details.
5.2.1 Geometry
!
The geometry of the footbridge is the foundation to modeling the structure and also the first step to do in LUSAS. In this stage, how to represent characteristics of footbridge needs to be considered very carefully.
The main structural system is composed by two edge beams, bridge deck, I-shaped girder and lateral bracings. The pavement is modeled as distributed mass and added to the bridge deck. In the same way, the parapet is also added to the edge beam as distributed mass. The lateral bracings are located at the support and within each span. In each span, the distance between lateral bracings is assumed to be constant. The geometric model in LUSAS without attributes is shown in Fig.5.4 below.
5.2.$FE$MODEL! ! 27!
Figure 5.4: Geometric model in LUSAS without attributes
5.2.2 Mesh
!
In order to avoid torsional effect on footbridge and consider shear deformation in the LUSAS model, all the structural elements of the bridge were modeled as thick shell and thick beam elements. The explanations of why choosing these two types of elements are discussed later.
Bridge girder, Stiffeners and Edge beam
A basic principle in building geometry of structure in LUSAS is that the designer should try to avoid creating unnecessary geometric lines as much as possible. Based on this principle, one way to mesh the girder is choosing the two middle lines of the bridge deck along the longitudinal direction of the bridge and meshing them as thick beam element. However, in this way, there are no available nodes for adding supports. In order to solve this problem, the bridge girder is firstly divided into three parts (top flange, web, and bottom flange) and then each part is meshed respectively. Specifically speaking, both the top flange and bottom flange are meshed with beam element while the web of girder is meshed with thick shell element. Alternatively, the whole girder can be meshed as beam element and connected with supports by very stiff beam. In this thesis, the former solution that meshing different parts of bridge girder respectively is adopted.
For bracings and edge beams, they are also meshed with thick beam element.
Bridge deck
The bridge deck is created as a shell structure in geometry and the influence of shear deformation cannot be neglected. By taking all these aspects into account, the 3D thick shell element is recommended to use.
28! ! CHAPTER(5.(THE(FOOTBRIDGE!
Convergence Test
After the mesh types are selected, the next step is to determine the mesh size. In order to find the most suitable mesh size, a convergence test is a necessity. The convergence test is carried out based on bending moment along longitudinal direction of footbridge with 3 different mesh sizes (0.5m, 0.4m, and 0.3m respectively). The results from LUSAS are shown in Fig.5.5. It should be noted that the values indicate the moment per unit width instead of total moment under the action of self-weight.
Figure 5.5: Result of convergence test (Unit:kNm)
According to Fig.5.5, it is obvious that the results from a element size of 0.5m already achieves good convergence. In order to save computation time, a mesh size of 0.5m is used for the whole structure. The summary of mesh information for all parts of the footbridge is listed in Table 5.1.
Table 5.1. Summary of mesh type and size
Member Element
name Structural
element name Dimension Interpolation
order Element
size Edge beams&
stiffeners& flanges of girder
BMI21 Thick beam 3D Linear 0.5m
Bridge deck& web of girder
QTS4 Thick shell 3D Linear 0.5m
The convergence test is also performed based on the calculated eigenfrequencies listed in Table 5.2 from which there are only slightly differences between results from different mesh sizes (d).
This proves the conclusion that the mesh size of 0.5m is enough for dynamic analyses.
5.2.$FE$MODEL! ! 29!
Table 5.2. Convergence in terms of eigenfrequencies
!
Eigenfrequency
Mode d=0.5m d=0.4m d=0.3m
1 2.742 Hz 2.734 Hz 2.735 Hz
2 2.879 Hz 2.877 Hz 2.875 Hz
3 3.263 Hz 3.262 Hz 3.260 Hz
4 3.599 Hz 3.596 Hz 3.592 Hz
5 3.912 Hz 3.909 Hz 3.906 Hz
6 4.206 Hz 4.203 Hz 4.199 Hz
7 4.523 Hz 4.519 Hz 4.514 Hz
8 4.613 Hz 4.611 Hz 4.609 Hz
9 4.795 Hz 4.784 Hz 4.773 Hz
10 4.957 Hz 4.952Hz 4.946 Hz
5.2.3 Material
!
After meshing is completed, the material properties should be defined and assigned to the corresponding structural elements. All structural elements in the FE model, except edge beams and bridge deck, are made of steel S355. The material properties of S355 can be found in EN 1993-1-1 [16] and listed in Table 5.3.
Table 5.3. Material properties of S355
Steel: S355
Young’s modulus 210GPa
Poisson’s ratio 0.3
Density 7.849*103kg/m3
Thermal expansion 12*10-6
Yield strength fy 355MPa
When it comes to edge beams and bridge deck, they are assumed to be made of concrete C40/50 with water to cement ratio less than 0.4. The material properties of C40/50 are listed in Table 5.4 based on EN 1992-1-1[17].
